Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals

Some of the most exotic condensed matter phases, such as twist grain boundary and blue phases in liquid crystals and Abrikosov phases in superconductors, contain arrays of topological defects in their ground state. Comprised of a triangular lattice of double-twist tubes of magnetization, the so-called 'A-phase' in chiral magnets is an example of a thermodynamically stable phase with topologically nontrivial solitonic field configurations referred to as two-dimensional skyrmions, or baby-skyrmions. Here we report that three-dimensional skyrmions in the form of double-twist tori called 'hopfions', or 'torons' when accompanied by additional self-compensating defects, self-assemble into periodic arrays and linear chains that exhibit electrostriction. In confined chiral nematic liquid crystals, this self-assembly is similar to that of liquid crystal colloids and originates from long-range elastic interactions between particle-like skyrmionic torus knots of molecular alignment field, which can be tuned from isotropic repulsive to weakly or highly anisotropic attractive by low-voltage electric fields.